A Quantitative Estimate on Fixed-Points of Composite Meromorphic Functions

1995 ◽  
Vol 38 (4) ◽  
pp. 490-495 ◽  
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Fixed Points ◽  
Lower Order ◽  
Quantitative Estimate ◽  
Meromorphic Functions ◽  
Transcendental Entire Function ◽  
Logarithmic Density ◽  
Image Position ◽  
Finite Lower Order

AbstractLet ƒ(z) be a transcendental meromorphic function of finite order, g(z) a transcendental entire function of finite lower order and let α(z) be a non-constant meromorphic function with T(r, α) = S(r,g). As an extension of the main result of [7], we prove thatwhere J has a positive lower logarithmic density.

scholarly journals ON SOLUTIONS TO NONHOMOGENEOUS ALGEBRAIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

2014 ◽  
Vol 97 (3) ◽  
pp. 391-403 ◽  
Author(s):  
LIANG-WEN LIAO ◽  
ZHUAN YE
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Meromorphic Solution ◽  
Differential Polynomial ◽  
Transcendental Entire Function ◽  
Algebraic Differential Equation ◽  
Algebraic Differential Equations ◽  
Image Position

AbstractWe consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then $$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$ With this in hand, we also prove that if $f$ is a transcendental entire function, then $f'p_k(f)+q_m(f)$ assumes every complex number $\alpha $, with one possible exception, infinitely many times, where $p_k(f), q_m(f)$ are polynomials in $f$ with degrees $k$ and $m$ with $k\ge m+1$. This result generalizes a theorem originating from Hayman [‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2)70(2) (1959), 9–42].

scholarly journals Radial distributions of Julia sets of meromorphic functions

2006 ◽  
Vol 81 (3) ◽  
pp. 363-368 ◽  
Author(s):  
Ling Qiu ◽  
Shengjian Wu
Keyword(s):  
Meromorphic Function ◽  
Lower Order ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Deficient Value ◽  
Borel Exceptional Value ◽  
Exceptional Value ◽  
Finite Lower Order

AbstractWe consider a meromorphic function of finite lower order that has ∞ as its deficient value or as its Borel exceptional value. We prove that the set of limiting directions of its Julia set must have a definite range of measure.

scholarly journals Derivation of Some Results on the Generalized Relative Orders of Meromorphic Functions

2017 ◽  
Vol 55 (1) ◽  
pp. 51-61
Author(s):  
Sanjib Kumar Datta ◽  
Tanmay Biswas
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Lower Order ◽  
Meromorphic Functions ◽  
Relative Order

Abstract In this paper we intend to find out relative order (relative lower order) of a meromorphic function f with respect to another entire function g when generalized relative order (generalized relative lower order) of f and generalized relative order (generalized relative lower order) of g with respect to another entire function h are given.

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals Fixed Points of Difference Operator of Meromorphic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Zhaojun Wu ◽  
Hongyan Xu
Keyword(s):  
Meromorphic Function ◽  
Fixed Points ◽  
Meromorphic Functions ◽  
Difference Operator ◽  
Transcendental Meromorphic Function ◽  
Exact Difference ◽  
Borel Exceptional Values ◽  
Nevanlinna Deficiency

Letfbe a transcendental meromorphic function of order less than one. The authors prove that the exact differenceΔf=fz+1-fzhas infinitely many fixed points, ifa∈ℂand∞are Borel exceptional values (or Nevanlinna deficiency values) off. These results extend the related results obtained by Chen and Shon.

Meromorphic functions with one deficient value

1969 ◽  
Vol 10 (3-4) ◽  
pp. 355-358 ◽  
Author(s):  
S. M. Shah
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Zero Order ◽  
Regularity Conditions ◽  
Deficient Value ◽  
Exceptional Value ◽  
Image Position

Let f(z) be a meromorphic function and write Here N(r, a) and T(r, f) have their usual meanings (see [4], [5]) and 0 ≧ |a| ≧ ∞. If δ(a, f) > 0 then a is said to be an exceptional (or deficient) value in the sense of Nevanlinna (N.e.v.), and if Δ(a, f) > 0 then a is said to be an exceptional value in the sense of Varliron (V.e.v.). The Weierstrass p(z) function has no exceptional value N or V. Functions of zero order can have atmost one N.e.v. [4, p. 114], but may have more than one V.e.v. (see [6], [8]). In this note we consider functions satisfying some regularity conditions and having one and only one exceptional value V.

Entire and Meromorphic Functions with Asymptotically Prescribed Characteristic

1965 ◽  
Vol 17 ◽  
pp. 383-395 ◽  
Author(s):  
Albert Edrei ◽  
Wolfgang H. J. Fuchs
Keyword(s):  
Entire Function ◽  
Convex Functions ◽  
Meromorphic Functions ◽  
Image Position ◽  
Increasing Function ◽  
Logarithmically Convex Functions

If f(z) is a non-constant, entire function, then Hadamard's three-circles theorem asserts thatis a convex, increasing function of log r. Hence, by well-known properties of logarithmically convex functions,

scholarly journals Uniformly bounded components of normality

2007 ◽  
Vol 143 (1) ◽  
pp. 85-101
Author(s):  
XIAOLING WANG ◽  
WANG ZHOU
Keyword(s):  
Entire Function ◽  
Transcendental Entire Function ◽  
Fatou Set ◽  
Image Position ◽  
Uniformly Bounded

AbstractSuppose that f(z) is a transcendental entire function and that the Fatou set F(f)≠∅. Set and where the supremum supU is taken over all components of F(f). If B1(f)<∞ or B2(f)<∞, then we say F(f) is strongly uniformly bounded or uniformly bounded respectively. We show that, under some conditions, F(f) is (strongly) uniformly bounded.

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